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Question: A steel wire of cross-sectional area 0.5\({\text{m}}{{\text{m}}^{\text{2}}}\) is held between two fi...

A steel wire of cross-sectional area 0.5mm2{\text{m}}{{\text{m}}^{\text{2}}} is held between two fixed supports. If tension in the wire is negligible and it is just taut at 20 0C{}^{\text{0}}{\text{C}}, determine tension when temperature falls 0 0C{}^{\text{0}}{\text{C}}. Young's modulus of elasticity is 21×101121 \times {10^{11}} dyne/cm - 2{\text{c}}{{\text{m}}^{{\text{ - 2}}}}and coefficient of linear expansion is 12×10−612 \times {10^{ - 6}}per degree centigrade. Assume that the distance between the supports remains the same.

Explanation

Solution

Hint- To solve this question, we use the basic theory of Thermal expansion. As we know Thermal expansion occurs when an object expands and becomes larger due to a change in the object's temperature. Similarly, in this case it will affect the operation of steel wire when temperature changes to 200C{\text{20}}{}^{\text{0}}{\text{C}}. Some basic formulas are used to get our desired result in this problem.

Formula used- ⇒Strain=ΔLL=αΔT \Rightarrow Strain = \dfrac{{\Delta L}}{L} = \alpha \Delta T
Stress = Y × Strain
Given: Y=21×1011dyn/cm221 \times {10^{11}}dyn/c{m^2}
α=12×10−6/0C\alpha = 12 \times {10^{ - 6}}/{}^0C
Change in temperature ΔT=200C\Delta T = 20{}^0C
Change in temperature A=0.5mm2=0.005cm2A = 0.5m{m^2} = 0.005c{m^2}

Complete answer:
Let the initial length of steel wire be L and let L be the steel wire's length when the temperature is reduced to00C0{}^0C.
Decrease in length due to compression,ΔL=LI−L\Delta L = {L^I} - L
UsingΔL=LαΔT\Delta L = L\alpha \Delta T
⇒Strain=ΔLL=αΔT=12×10−6×20=2.4×10−4\Rightarrow Strain = \dfrac{{\Delta L}}{L} = \alpha \Delta T = 12 \times {10^{ - 6}} \times 20 = 2.4 \times {10^{ - 4}}
From Hooke's law, Stress=Y×Strain
⟹ Tension T=Y×Strain × A
∴T=21×1011×2.4×10−4×0.005=252×104dyn\therefore T = 21 \times {10^{11}} \times 2.4 \times {10^{ - 4}} \times 0.005 = 252 \times {10^4}dyn
⟹ T=25.2N (1N=104dyn{10^4}dyn)
Therefore, tension of the steel wire is 25.2 N.

Note- The expansion can occur in length of iron pendulum in which case it is called Linear Expansion. And If we take a square tile and then after heat it, the expansion will be on two fronts that is length and breadth, and it is called Area Expansion. Similarly, if we take a cube shape structure and heat it, all its sides expand and now the body experiences an increase in the overall volume of the structure and it is called Volume Expansion.